In Fig.REF , we show the behavior of \(\mu /\mu _{max} -C/C_{max}\)  with respect to the constant parameters of the Taub NUT black hole in \(\hat{n}=0.05, 0.1, 0.24, 0.25\) . In such a case, we face with three cases: i) For \(C=\Omega C_{max}\) , \(\mu \)  becomes zero. ii) For \(C>\Omega C_{max}\)  in \(\hat{n}\le 0.24\) , the free energy becomes zero so that the microscopic degrees of freedom are repulsive. iii) For \(C<\Omega C_{max}\) , the degrees of freedom are attractive.
Also, in \(\hat{n}=0\) , we reproduce the results of the RN-AdS and Kerr-AdS black holes [1]}.
